Dual Zariski Spaces of Modules

Abstract

Let [Formula: see text] be a commutative ring with identity, [Formula: see text] be an [Formula: see text]-module, [Formula: see text] denote the set of all submodules of [Formula: see text] and [Formula: see text]. For any submodule [Formula: see text] of [Formula: see text], we set [Formula: see text] and [Formula: see text]. Consider [Formula: see text], where [Formula: see text] is the set of all ideals of [Formula: see text]. We set [Formula: see text] and [Formula: see text] for any ideal [Formula: see text] of [Formula: see text]. In this paper, we investigate when, for arbitrary [Formula: see text] and [Formula: see text] as above, [Formula: see text] and [Formula: see text] form a topology and a semimodule, respectively. We investigate the structure of [Formula: see text] in the case that it is a semimodule.